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Binary independent and dependent variables
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Binary independent variables
y = 0 + 1x1 + 2x2 + . . . kxk + u
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Dummy Variables
A dummy variable is a variable that takes on the value 1 or 0
Examples: male (= 1 if are male, 0 otherwise), NYSE (= 1 if stock listed in NYSE, 0 otherwise), etc.
Dummy variables are also called binary variables
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A Dummy Independent Variable
Consider a simple model with one continuous variable (x) and one dummy (d)
y = 0 + 0d + 1x + u
This can be interpreted as an intercept shift
If d = 0, then y = 0 + 1x + u
If d = 1, then y = (0 + 0) + 1x + u
The case of d = 0 is the base group
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Example,
Y=turnover=trading volume/outstanding shares
X=size of firms=log (stock price*outstanding shares)
D=1 for NYSE stocks, 0 otherwise (AMEX and NASDAQ)
If you assume the slope is the same for both groups, then regress y = 0 + 0d + 1x + u
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Example of 0 > 0
x
y
{0
}0
y = (0 + 0) + 1x
y = 0 + 1x
slope = 1
d = 0
d = 1
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Dummies for Multiple Categories
We can use dummy variables to deal with something with multiple categories Suppose everyone in your data is either a HS dropout, HS grad only, or college grad To compare HS and college grads to HS dropouts, include 2 dummy variables hsgrad = 1 if HS grad only, 0 otherwise; and colgrad = 1 if college grad, 0 otherwise
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Multiple Categories (cont)
Any categorical variable can be turned into a set of dummy variables
The base group is represented by the intercept; if there are n categories there should be n – 1 dummy variables
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Example,
Y=turnover=trading volume/outstanding sharesX=size of firms=log (stock price*outstanding shares)
D=1 for NYSE stocks, 0 otherwise (AMEX and NASDAQ)If you assume the slope and intercept are different for both groups, you could run two separate regression for each group, or…
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Interactions with Dummies
Can also consider interacting a dummy variable, d, with a continuous variable, x
y = 0 + 1d + 1x + 2d*x + u
If d = 0, then y = 0 + 1x + u
If d = 1, then y = (0 + 1) + (1+ 2) x + u
This is interpreted as a change in the slope (and intercept).
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y
x
y = 0 + 1x
y = (0 + 0) + (1 + 1) x
Example of 0 > 0 and 1 < 0
d = 1
d = 0
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Caveats
A typical use of a dummy variable is when we are looking for a program effect
For example, we may have individuals that received job training and wish to test the effect of training.
We need to remember that usually individuals choose whether to participate in a program, which may lead to a self-selection problem
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Self-selection Problems
If we can control for everything that is correlated with both participation and the outcome of interest then it’s not a problem
Often, though, there are unobservables that are correlated with participation. For example, people’s skill before the training. Low skill people may be more likely to take the training.
In this case, the estimate of the program effect is biased.
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Self Selection Problem: An example
Suppose one professor finds that his evening class students perform better that his daytime class.Does that mean the professor teaches better at night or students learn better at night?No. It turns out that students participating in night classes might be more hard working. They care to come at night, sometimes after a daytime work, etc.
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Self-selection Problems
If we can control (control means adding more explanatory variables) for everything that is correlated with both participation (the dummy variable) and the outcome (y), then it’s not a problem.
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Binary Dependent Variables Logit and Probit models
P(y = 1|x) = G(0 + x)
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Binary Dependent Variables
A linear probability model can be written as P(y = 1|x) = 0 + x
A drawback to the linear probability model is that predicted values are not constrained to be between 0 and 1
An alternative is to model the probability as a function, G(0 + x), where 0<G(z)<1
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The intuition
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The Probit Model
One choice for G(z) is the standard normal cumulative distribution function (cdf) G(z) = (z) ≡ ∫(v)dv, where (z) is the standard normal, so (z) = (2)-1/2exp(-z2/2) This case is referred to as a probit model Since it is a nonlinear model, it cannot be estimated by our usual methods Use maximum likelihood estimation
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The Logit Model
Another common choice for G(z) is the logistic function
G(z) = exp(z)/[1 + exp(z)] = (z)
This case is referred to as a logit model, or sometimes as a logistic regression
Both functions have similar shapes – they are increasing in z, most quickly around 0
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Probits and Logits
Both the probit and logit are nonlinear and require maximum likelihood estimation
No real reason to prefer one over the other
Traditionally saw more of the logit, mainly because the logistic function leads to a more easily computed model
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Interpretation of Probits and Logits
In general we care about the effect of x on P(y = 1|x), that is, we care about ∂p/ ∂x
For the linear case, this is easily computed as the coefficient on x
For the nonlinear probit and logit models, it’s more complicated:
∂p/ ∂xj = g(0 +x)j, where g(z) is dG/dz
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For logit model
∂p/ ∂xj =j exp(0 +x(1 + exp(0 +x) ) 2
For Probit model:
∂p/ ∂xj = j (0 +x) = j (2)-1/2exp(-(0 +x2/2)
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Interpretation (continued)
Can examine the sign and significance (based on a standard t test) of coefficients,
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